Complex Numbers Have More Uses Than You Think
ฝัง
- เผยแพร่เมื่อ 12 มิ.ย. 2024
- Complex numbers are often seen as a mysterious or "advanced" number system mainly used for solving similarly mysterious or "advanced" problems. But really, once you get used to them, they're really an elegant and (ironically) simple mathematical tool with application to more down-to-earth problems besides Quantum Mechanics or advanced Differential Equations or something. Let's see what these numbers can do!
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=Chapters=
0:00 - Intro
1:45 - Complex number basics
3:31 - Interpreting complex number multiplication
7:55 - Angular velocity
10:42 - Calculating angular velocity using complex numbers
15:18 - Interpreting the formula
17:59 - More uses of complex numbers
19:48 - Special announcement!
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CREDITS
The music tracks used in this video are (in order of first appearance): "Dream Escape", "Checkmate", "Orient", "Rubix Cube", "Frozen in Love"
The track "Rubix Cube" comes courtesy of Audionautix.com
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The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:
github.com/morpho-matters/mor...
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"Complex Numbers are the language of 2D rotation" 7:54
My friend once asked for applications of imaginary numbers. My Dad (an Engineer) said, "They're just for rotation, aren't they?". I couldn't believe that none of my Maths Professors had ever put it that bluntly!
Probably because it's not true in a mathematics context that complex numbers are just for rotation
@@regarrzo The best suggestion I'd been able to make was that they showed in the analysis of a system's stability? Imaginary eigenvalues indicated oscillation, if I recall?
@@dylanparker130 I don't really know what is meant by asking for applications. Is your friend looking for an engineering/science perspective or a mathematical perspective?
In science and engineering, imaginary numbers can simplify many calculations dealing with perioid things. In mathematics, they are interesting because of their properties alone, e.g. being an algebraically closed field, holomorphic functions being infinitely differentiable, ...
I don't really understand what you mean with your comment. What kind of system are you referring to? Linear systems with matrix with imaginary eigenvalues?
@@regarrzo I was referring to systems with equilibria whose stability can be studied through the eigenvalues of an associated Jacobian Matrix.
@@dylanparker130 Ahh, then I understand. Thanks for clearing it up!
I would love to see a video on quaternions
Me too! I love them, but I don't understand the polar-form.
same because i don’t understand them at all
Probably a series of videos
Also would like to see a video on quaternions
Strange we'd get this video before quaternions, given how widespread they are in applications.
the animations are so clean that I almost forgot I was watching a math video I was so mesmerized 😭
Complex Analysis is such an interesting field, and I think everyone would love to see more on this topic. Great video!
I concur. Let's analyze this complex subject.
I found a really excellent lecture playlist that covers the most important parts. th-cam.com/play/PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg.html
The formula f'(x)/f(x) is called the logarithmic derivative, because it is also equal to the derivative of log(f(x)). It can be interpreted as a proportional rate of change. For example, a value that grows by a constant 10% per year has a constant logarithmic derivative, and the original function is an exponential. It is then interesting that this same formula appears for angular speed as well, though I think it makes intuitive sense if you think about it, since angular speed is the scale-invariant form of circular speed. The real part of the formula in the video should also corresponds to the proportional rate of change in the magnitude of f(x), so then we have a complete interpretation of the complex valued f'(x)/f(x) as encoding both the angular velocity and the growth rate of the magnitude.
wow
In this example, we get that the w'= Im[f'(z)/f(z)]. How would you write this out though? The only way i can think to do it is taking the derivative of cos(t)+isin(t) and using that for the numerator, but doesn't that still end up suffering from the discontinuity problem he mentioned?
Hi, Morph. This is a really great video!
I also appreciate how you include well-written captions. Not every math channel does that.
I hope you'll cover geometric algebra (Clifford Algebra) together with quaternions! Would be fun seeing them related and recover all this geometrical intuition in a single framework.
I've just been independently studying geometric algebra (blame Marc Ten Bosch literally dissing quaternions starting me down that rabbit hole) and Grassmann numbers/algebra (because of spinors in QM) only to find out that they come together at Clifford Algebra, so I should ought to learn about that whole situation next. 😁
Seconded. It's really cool to see how objects satisfying the axioms of quaternions arise out of geometric algebra. Gives them context. By themselves quaternions are rather mysterious and you have to wave your hands a lot to justify using four-dimensional objects to manipulate 3D coordinates.
@@lumipakkanen3510
Quaternions being equivalent to 3D rotors is really not all that a useful insight for practical applications, in fact it only causes confusion.
The sooner everyone outside of pure maths forgets about quaternions the better, geometric algebra is a much better framework.
@@viliml2763 True from a fresh perspective. However we now have a history of using quaternions in 3D modeling, so bridging the gap is in order. There are also low-level arguments for using quaternions internally to save a few float multiplications even if the user interface speaks GA. Also remember that quaternions are a geometric algebra in their own right.
Just finished an intro complex analysis class at uni last semester, and I gotta say this is a really good way to explain this stuff. Kind of sad that Cauchy's Integral Formula didn't show up here, especially because it's related to the rotational velocity problem, but I understand why that might be a bit in-depth for a 20 minute video that already needs to spend most of its time explaining the rotational velocity problem.
i'm super rusty on my calculus... but the geometric interpretation afterwards is just *so* intuitive and brilliant! Thank you for making this video, and for all the others. ❤
I've been looking everywhere for uses of complex numbers for the single most important paper on my entire education. It's due in 3 days, and you sir, just saved my life. Awesome video!
Hand down the best explanation of complex arithmetic I’ve ever seen! Thanks for the video!
I just want to say that your videos are one of the most interesting thing in math TH-cam.
Your visuals are excellent and so helpful. Motivation is so important to learning math and you have hit the nail on the head with this video. Thank you!
Love these videos. So easy to understand and very informative. Can't wait for more to come
I love complex numbers!
Same
Have seggsss
@@ujjawalk6780 Should I have sex with complex numbers? I think it's going to take forever because there are so many of them.
I love undertime slopper!
@@FunnyAndCleverHandle What's that?`Is that a guy on Tiktok?
Love your videos! First time catching one on release
I've loved every one of your videos so far, and I'm excited to see where you take the channel in the future! I wish I was in a position where I could join your patreon, perhaps someday. In the meantime, keep up the great work!
Thank you so much!! As a senior in high school who is looking into studying maths and physics at university, your videos are an invaluable asset for sparking my curiosity and building my intuition for mathematics.
Amazing. Simply amazing and elegant presentation of this mathematical field. Keep on the excellent work!
19:21 I would love to see a video about quaternions from you in the future! I loved this one!
This video could not have been more timely for me, thank you Morphocular :D
Morphocular, your topics and videos are always so great! Thanks so much for the work you put into them!
I can't help but add, for anyone interested in the Riemann Zeta function and its mythical nontrivial zeros and understanding how to find them, the mentions of polar parametric functions and epicycles at the end of the video are incredibly useful. Just take a peek at the Dirichlet Eta Function and its amazing relationship with the Riemann Zeta function. ^_^
While the "mysterious" angle formula arctan is indeed not continuous, the derivative actually is and yields the same result after short calculation:
Θ' = (x y' - y x') / (x² + y²)
No imaginary numbers needed, but the visual presentation is still worthy of a gold medal
Um what about theta = 0?
for Θ=0 => y=0, regardless from which side you approach the x-axis
so Θ' = y' / x
which is the correct result
I do look forward to quaternions also. Your unique viewpoint helped me to see more. Thks
Great video! The awesome visualizations helped me understand complex numbers a lot more 😃
WOW!!😍 You increased my affection towards "complex" numbers....though I like to call them "Frisky numbers" ....I personally find them pretty interesting like they play around in the plane like child🥰
keep it up 👍
Great video! As a physics student with a passion for maths, this was really interesting and useful to watch.
This is so beautiful , thank you so much! I am so grateful
this hwas awesome, i didn't know that you could find angular velocity like this.
i hope for another great video explaning quarternions and maybe also a video on others like the split-complex numbers and tessarines
Without teachings like this, found both on the internet and in good books, I would not be able study science. I am completely unable to learn by having a bunch of seemingly meaningless information being thrown at my face.
Thanks a ton for sharing
Really cool video and well done on the channel explosion, I would really love to see how geometric algebra can explain rotations in not only three but n dimensions, multi vectors ftw
loved it dude!
keep it up
greetings from brazil
OOOOOOOH, I'd love a vide from you on quarternions! I loved the ones from Numberphile and 3b1b, but i think your beautiful visualizations and skill for revealing intuition will be a great addition to the topic
Thanks very much for making this video. I didn't know that interpretation of multiplication by a complex number! it sounds a lot like the spectral decomposition of a matrix.
Even though it has been decades I touched or used mathematics. It facinates me to revisit the fundamentals of mathematics for a new perspective just for pure joy and appreciation of mathematics, which I feel I could not do justice a teenage student.
Your video very elegantly explains it... Thanks for making such useful videos.
Wow! Top notch content. Cannot wait to watch the quaternion video.
Genius. Thank you. I find a great value in your videos
Loved your video. I just have started Learning complex numbers in high school and getting to learn so much about it made me mad curious to learn more about it .
First time I fully understood this topic. One of the most useful vídeos for me in internet.
love how you put the background in a dimmed yellow, so my eyes won't get tired
Great video. Love to see more on complex numbers, Fourier, epicycles, and quaternions 3D rotations…and General Stokes differential forms if you are into that. Thank you!
Could checkout 3blue1brown's video on quaternions
Thanks for making the background audio stand back a little and not dominate your voiceover
Thank you very much for reaching!
Another great video!
Thanks! Please do the quaternion time derivative.
I like this video
Makes me excited to learn more about it in my next semester
Awesome video, neat explanation
I loved that step at 12:00 - genius!
Great video, it's nice to see a more original video introducing complex numbers rather than regurgitating the rules. I feel like those who like this video would also like "Are Complex Numbers Forced Upon Us? Multiplication in High Dimensions" by James Tanton, it shows their elegance nicely imo
best video on complex number for understanding its practical use! best
very inspiring video. Thank you for the masterpiece.
your knowledge and experience help to understand a lot. appreciate a lot. kindly make such beautiful videos. we will also support from our side as much we can as students.
I used to want to extend every new thing I learned about complex numbers to quaternions. A few years ago when learning about how quaternions are useful for 3D rotations and more efficient than matrix rotations, I stumbled into geometric algebra.
Now I need to know how everything I learn about complex numbers extend to geometric algebras!
Fun fact is that complex numbers, quaternions, and vectors, and a bunch or hyper complex number systems are all subalgebras of geometric algebras.
Plus other geometric numbers square to 1 and 0 turning circular rotation into hyperbolic rotation or translation. And they operate on any number of dimensions, not just 2 or 3.
this is good work, keep going!
Very well done. Keep up the spirit.
Beautiful video 🙂
Your videos are sooooooooooo USEFUL! I know you Will say thank you
Excellent stuff.
Thank you. Hope to see your quaternion video.
Excellent video on this topic, this also explains how rotation matrix works in computer graphics
Thank you.
As someone who just completed a secondary school maths curriculum, these videos are perfect since I have just the right amount of prerequisite knowledge to understand what is meant by these videos
Amazing
your channel is so underrated
Amazing work, I don't know what to say. I really, really appreciate it.
Great video as always.
Also, was the "angle" at 1:30 an intended pun?
Loved and Subscribed from India
Amazing, continue this exelent channel
I finally understand this video!! Dope
very nice video, hope to see some explanation about quaternions
Love your videos.
Great video!
You’re as intelligent as you’re kind to us, it’s pleasure to be part of the journey of this channel
Please continue. Great things from small.
I'll love it if you ever make a video like this on quaternions!
dude! I wish I would've came across this video before Signals and Systems class, I could've gotten a better grade! dang! It's sooo good, this 20 min video would've made an entire semester easier.
took a couple minutes, but i got it, and its absolutely elegant af
and intuitive
I'm was in awe the whole time 😭
Thank you sir 👍
love your videos!
You are amazing, thank you
Great video! :D
I saw the last part of your video with the future topics list. Please do the calculus of variations. There aren’t enough good videos on the topic.
Videos left: hyperbolic numbers
quaternions
biquaternions
octonions
split-octonions
sedenions
trigintaduonions
and dual numbers.
cool
Nice to watch
please don't stop making these videos
graet explanation thanks
Stop making me excited for learning calc! Just one more year before it begins. Also love the animations and how these topics always tie up in the end
Thanks!
i can be rewritten as the product of the x and y basis vector, defined such that xy=-yx, x^2=1, and y^2=1
multiplying vectors by i has the same effect as multiplying a complex number by i
for example to rotate 2x+3y a quarter turn, we can do (2x+3y)xy=2xxy+3yxy=2y-3xyy=-3x+2y
it gives a nice geometric interpretation of i as a plane (bivector)
Wowwowwow, this is really good stuff. I'm in teh first year of my bachelor's studies so I was about to close the video cause it started from stuff I already knew, but man am I glad I just skipped to 10 minutes cause that trick is so cool. I can't believe that I hadn't seen this before.
Can't wait for a video on Quaternions
The new thumbnail is much better
I think I found one of my new favorite math fields
Sir, your explanations are pretty, really nice. You explain in a very clear way. I can imagine how long it takes for you to produce a video like this. Congratulations Friend, for your effort. I am an observer [economist] from Brazil! I do not know where you are!
I felt a lot better about complex numbers after I took my first complex analysis course. They're really second nature to me now, and I just view them as the plane with a neat multiplication rather than something spooky and mysterious
thank you sir.
this is sick!!
Please make a video (series?) on calculus of variations. This is a wide open hole that hasn’t really been covered yet on TH-cam to my knowledge
Beautiful, lucid. Similar to another math explaner in format, but without the affectation and twee.
I personally like how you can make Acos, Atan from complex numbers and log, since a number x = r * exp(it) if r is 1, then log(x)/i=t the angle of x. cos(t)=x is the length in horizontal direction of a circle of radius 1, so it is the angle of x+i*sqrt(1-x^ 2) or Acos(x) = log(x+i*sqrt(1-x^ 2))/i. In the same way Atan is the angle from something of horisontal direction 1 and height x. However 1+i*x does not have length 1, but 1-i*x have opposite angle but same length, so Atan(x) = log((1+i*x)/(1-i*x))/(2i), since the ratio have length 1, and when you divide, the angle gets a minus, so you get twice the angle.
16:50 actually blew my mind